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176. CMB 2000 (vol 43 pp. 208)

Matoušková, Eva
 Extensions of Continuous and Lipschitz Functions We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$. Keywords:extension, continous, Lipschitz, Banach spaceCategories:54C20, 46B10

177. CMB 2000 (vol 43 pp. 138)

Boyd, C.
 Exponential Laws for the Nachbin Ported Topology We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet spaces $E$ and $F$ that we have the topological identity $$\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H} \Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega \biggr).$$ Analogous results for the compact open topology have long been established. We also give an example to show that the (Qno) hypothesis on both $E$ and $F$ is necessary. Categories:46G20, 18D15, 46M05

178. CMB 2000 (vol 43 pp. 69)

Kaminker, Jerome; Perera, Vicumpriya
 Type II Spectral Flow and the Eta Invariant The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type~I and Type~II spectral flow. Categories:19K56, 46L80

179. CMB 1999 (vol 42 pp. 344)

Koldobsky, Alexander
 Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes We define embedding of an $n$-dimensional normed space into $L_{-p}$, $0 Categories:42A82, 46B04, 46F12, 60E07 180. CMB 1999 (vol 42 pp. 321) Kikuchi, Masato  Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established. Keywords:martingale inequalities, rearrangement invariant function spacesCategories:60G44, 60G46, 46E30 181. CMB 1999 (vol 42 pp. 274) Dădărlat, Marius; Eilers, Søren  The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite, real rank zero$C^\ast$-algebras$E$and$E'$for which there is an isomorphism of ordered groups$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$which is compatible with all the coefficient transformations. The$C^\ast$-algebras$E$and$E'$are not isomorphic since there is no$\Theta$as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra$\OO_\infty$one obtains a pair of non-isomorphic, real rank zero, purely infinite$C^\ast$-algebras with similar properties. Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps,$C^\ast$-algebras, real rank zero, purely infinite, classificationCategories:46L35, 46L80, 19K14 182. CMB 1999 (vol 42 pp. 139) Bonet, José; Domański, Paweł; Lindström, Mikael  Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Every weakly compact composition operator between weighted Banach spaces$H_v^{\infty}$of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space$H_v^{\infty}$are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces. Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operatorCategories:47B38, 30D55, 46E15 183. CMB 1999 (vol 42 pp. 221) Liu, Peide; Saksman, Eero; Tylli, Hans-Olav  Boundedness of the$q$-Mean-Square Operator on Vector-Valued Analytic Martingales We study boundedness properties of the$q$-mean-square operator$S^{(q)}$on$E$-valued analytic martingales, where$E$is a complex quasi-Banach space and$2 \leq q < \infty$. We establish that a.s. finiteness of$S^{(q)}$for every bounded$E$-valued analytic martingale implies strong$(p,p)$-type estimates for$S^{(q)}$and all$p\in (0,\infty)$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function$S^{(q)}$) of the complex spaces$E$that admit an equivalent$q$-uniformly PL-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the$L^p$-boundedness of the usual square-function on scalar-valued analytic martingales. Categories:46B20, 60G46 184. CMB 1999 (vol 42 pp. 118) Rao, T. S. S. R. K.  Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space$\WC(K,X)^\ast$For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions,$M$-idealsCategories:46B20, 46E40 185. CMB 1999 (vol 42 pp. 104) Nikolskaia, Ludmila  InstabilitÃ© de vecteurs propres d'opÃ©rateurs linÃ©aires We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors$(x_n)$to be eigenvectors$Tx_n= \lambda_n x_n$($\lambda_n \noteq \lambda_k$for$n\noteq k$) of a bounded operator$T$(admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1)~for the sequence$(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$to be admissible for every admissible$(x_n)$and for a suitable choice of small numbers$\epsilon_n\noteq 0$it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist$\gamma_n\in \C$such that$v_n= \gamma_n v_{k}$for$n\geq k$(Theorem~2); (2)~for a bounded operator$A$to transform admissible families$(x_n)$into admissible families$(Ax_n)$it is necessary and sufficient that$A$be left invertible (Theorem~4). Keywords:eigenvectors, minimal families, reproducing kernelsCategories:47A10, 46B15 186. CMB 1998 (vol 41 pp. 279) Acosta, María D.; Galán, Manuel Ruiz  New characterizations of the reflexivity in terms of the set of norm attaining functionals As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space$X$is very smooth or its bidual satisfies the$w^{\ast }$-Mazur intersection property, then either$X$is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if$X$has the Mazur intersection property and so, if the norm of$X$is Fr\'{e}chet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion. Categories:46B04, 46B10, 46B20 187. CMB 1998 (vol 41 pp. 257) Bagby, Thomas; Gauthier, P. M.  Note on the support of Sobolev functions We prove a topological restriction on the support of Sobolev functions. Categories:46E35, 31B05 188. CMB 1998 (vol 41 pp. 145) Fry, R.  Smooth partitions of unity on Banach spaces It is shown that if a Banach space$X$admits a$C^k$-smooth bump function, and$X^{*}$is Asplund, then$X$admits$C^k$-smooth partitions of unity. Category:46B20 189. CMB 1998 (vol 41 pp. 240) Xia, Jingbo  On certain$K$-groups associated with minimal flows It is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial$K_1$-group. We show in this note that if the unique ergodicity is dropped, then such$K_1$-group can be non-trivial. Therefore, in the general setting of minimal flows, even the$K$-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal. Categories:46L80, 47B35, 47C15 190. CMB 1998 (vol 41 pp. 225) Vanderwerff, Jon  Mazur intersection properties for compact and weakly compact convex sets Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively. Categories:46B03, 46B20, 46A55 191. CMB 1998 (vol 41 pp. 41) Giner, E.  On the Clarke subdifferential of an integral functional on$L_p$,$1\leq p < \infty$Given an integral functional defined on$L_p$,$1 \leq p <\infty$, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand. Keywords:Integral functional, integrand, epi-derivativeCategories:28A25, 49J52, 46E30 192. CMB 1997 (vol 40 pp. 443) Hadwin, Don; Orhon, Mehmet  Reflective Representations and Banach C*-Modules Suppose${\cal A}$is a unital$C$*-algebra and$m\colon{\cal A}\to B(X)$Categories:47D30, 46L99 193. CMB 1997 (vol 40 pp. 488) Maouche, Abdelaziz  CaractÃ©risations spectrales du radical et du socle d'une paire de jordan-banach If$f$and$g$are two analytic functions from a domain$D$of the complex plane into respectively the Banach spaces$V^+$and$V^-$, we prove that$\lambda\mapsto \Sp\bigl(f(\lambda),g(\lambda)\bigr)$is an analytic multivalued function. From this derives the subharmonicity of the functions$\lambda\mapsto \rho_V\bigl(f(\lambda),g(\lambda)\bigr)$and$\lambda\mapsto \log\rho_V\bigl(f(\lambda),g(\lambda)\bigr)$where$\rho$denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem. Keywords:Spectre, rayon spectral, multifonction analytique, quasi-inverse,, paire de Jordan-Banach, radical de Jacobson, socle.Categories:46H70, (17A15) 194. CMB 1997 (vol 40 pp. 356) Mazet, Pierre  Principe du maximum et lemme de Schwarz, a valeurs vectorielles Nous {\'e}tablissons un th{\'e}or{\e}me pour les fonctions holomorphes {\a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles. Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux.Categories:30C80, 32A30, 46G20, 52A07 195. CMB 1997 (vol 40 pp. 129) Badea, Catalin  Sur les caractÃ¨res d'une algÃ¨bre de Banach A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem. Categories:46H05, 32A15 196. CMB 1997 (vol 40 pp. 254) Saito, Kichi-Suke; Watatani, Yasuo  Subdiagonal algebras for subfactors II (finite dimensional case) We show that finite dimensional subfactors do not have subdiagonal algebras unless the Jones index is one. Categories:46K50, 46L37 197. CMB 1997 (vol 40 pp. 133) Blackmore, T. D.  Derivations from totally ordered semigroup algebras into their duals For a well-behaved measure$\mu$, on a locally compact totally ordered set$X$, with continuous part$\mu_c$, we make$L^p(X,\mu_c)$into a commutative Banach bimodule over the totally ordered semigroup algebra$L^p(X,\mu)$, in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from$L^p(X,\mu)$into its dual module and in particular shows that if$\mu_c$is not identically zero then$L^p(X,\mu)$is not weakly amenable. We show that all bounded derivations from$L^1(X,\mu)$into its dual module arise in this way and also describe all bounded derivations from$L^p(X,\mu)$into its dual for$1 Categories:43A20, 46M20

198. CMB 1997 (vol 40 pp. 183)

Kepert, Andrew G.
 The range of group algebra homomorphisms A characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P.~J.~Cohen. Categories:43A22, 22B10, 46J99

199. CMB 1997 (vol 40 pp. 10)

Borwein, Jon; Vanderwerff, Jon
 Convex functions on Banach spaces not containing $\ell_1$ There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions. Categories:46A55, 46B20, 52A41
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